You can absolutely prove a negative.
Given
1: A \Rightarrow B
2: \neg B
We can prove
3: \neg A
Maybe you meant that “you can’t prove non-existence”. But that’s also false: ‘square circles’ don’t exist, and their non-existence follows from the meaning of ‘square’ and ‘circle’.
So though you appeal to “logical philosophical training”, you’re really talking about empirical proof, not logical proof, e.g. “you can’t empirically prove that something doesn’t exist”. I still disagree.
First, empirical reasoning embeds a lot of logical reasoning: If we agree that a thing’s existence would entail certain observable facts about the world, and we agree that we do not observe those facts, then we are in the same case with which I began above: (A \Rightarrow B) \&( \neg B) \therefore \neg A
But we can also approach the question by acknowledging more general limits on empiricism. We can be Cartesian skeptics, and pretty effectively argue that we can’t even prove that a thing does exist, let alone that it doesn’t. And I’m happy to admit that I may be insane or deceived, that all my observations may be mistaken, and in that sense I cannot prove that ghosts don’t exist any more than I can prove that my hands exist.
But if we are slightly less skeptical, and we accept that we can observe the world, and talk about what we observe; if we accept that you and I could independently measure certain aspects of reality and our measurements would be in reasonable agreement; if, further, we can reason about how the world works, because we can independently observe how some part of it works and arrive at equivalent models for how it will work; if we take as a given that there is an objective reality to which we all have access – as we must for any kind of empiricism to work – that we can prove that certain things do not exist in much the same way as we can prove that they do.
I began this thread by specifying what specifically I’m saying does not exist. And I argued that we know that those things do not exist because their existence would entail things about the world that we do not observe. And if A entails B, and we observe that not B, then we’ve proven that not A: There Are No Ghosts.